315 research outputs found
Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns
We prove that the Stanley-Wilf limit of any layered permutation pattern of
length is at most , and that the Stanley-Wilf limit of the
pattern 1324 is at most 16. These bounds follow from a more general result
showing that a permutation avoiding a pattern of a special form is a merge of
two permutations, each of which avoids a smaller pattern. If the conjecture is
true that the maximum Stanley-Wilf limit for patterns of length is
attained by a layered pattern then this implies an upper bound of for
the Stanley-Wilf limit of any pattern of length .
We also conjecture that, for any , the set of 1324-avoiding
permutations with inversions contains at least as many permutations of
length as those of length . We show that if this is true then the
Stanley-Wilf limit for 1324 is at most
Generalized permutation patterns - a short survey
An occurrence of a classical pattern p in a permutation π is a subsequence of π whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidance—or the prescribed number of occurrences— of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns
Some open problems on permutation patterns
This is a brief survey of some open problems on permutation patterns, with an
emphasis on subjects not covered in the recent book by Kitaev, \emph{Patterns
in Permutations and words}. I first survey recent developments on the
enumeration and asymptotics of the pattern 1324, the last pattern of length 4
whose asymptotic growth is unknown, and related issues such as upper bounds for
the number of avoiders of any pattern of length for any given . Other
subjects treated are the M\"obius function, topological properties and other
algebraic aspects of the poset of permutations, ordered by containment, and
also the study of growth rates of permutation classes, which are containment
closed subsets of this poset.Comment: 20 pages. Related to upcoming talk at the British Combinatorial
Conference 2013. To appear in London Mathematical Society Lecture Note Serie
On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia
We construct a sequence of finite automata that accept subclasses of the
class of 4231-avoiding permutations. We thereby show that the Wilf-Stanley
limit for the class of 4231-avoiding permutations is bounded below by 9.35.
This bound shows that this class has the largest such limit among all classes
of permutations avoiding a single permutation of length 4 and refutes the
conjecture that the Wilf-Stanley limit of a class of permutations avoiding a
single permutation of length k cannot exceed (k-1)^2.Comment: Submitted to Advances in Applied Mathematic
On consecutive pattern-avoiding permutations of length 4, 5 and beyond
We review and extend what is known about the generating functions for
consecutive pattern-avoiding permutations of length 4, 5 and beyond, and their
asymptotic behaviour. There are respectively, seven length-4 and twenty-five
length-5 consecutive-Wilf classes. D-finite differential equations are known
for the reciprocal of the exponential generating functions for four of the
length-4 and eight of the length-5 classes. We give the solutions of some of
these ODEs. An unsolved functional equation is known for one more class of
length-4, length-5 and beyond. We give the solution of this functional
equation, and use it to show that the solution is not D-finite. For three
further length-5 c-Wilf classes we give recurrences for two and a
differential-functional equation for a third. For a fourth class we find a new
algebraic solution. We give a polynomial-time algorithm to generate the
coefficients of the generating functions which is faster than existing
algorithms, and use this to (a) calculate the asymptotics for all classes of
length 4 and length 5 to significantly greater precision than previously, and
(b) use these extended series to search, unsuccessfully, for D-finite solutions
for the unsolved classes, leading us to conjecture that the solutions are not
D-finite. We have also searched, unsuccessfully, for differentially algebraic
solutions.Comment: 23 pages, 2 figures (update of references, plus web link to
enumeration data). Minor update. Typos corrected. One additional referenc
The limit of a Stanley-Wilf sequence is not always rational, and layered patterns beat monotone patterns
We show the first known example for a pattern for which is not an integer. We find the exact value of the
limit and show that it is irrational. Then we generalize our results to an
infinite sequence of patterns. Finally, we provide further generalizations that
start explaining why certain patterns are easier to avoid than others. Finally,
we show that if is a layered pattern of length , then
holds.Comment: 10 pages, 1 figur
- …